On the strength of connectedness of a random hypergraph
Bollobás and Thomason (1985) proved that for each , with high probability, the random graph process, where edges are added to vertex set uniformly at random one after another, is such that the stopping time of having minimal degree is equal to the stopping time of becoming -(vertex-)connected. We extend this result to the -uniform random hypergraph process, where and are fixed. Consequently, for and , the probability that the random hypergraph models and are -connected tends to
Keywords: random hypergraph; vertex connectivity
Let denote the random -uniform hypergraph with vertex set , where each of the potential (hyper)edges of cardinality is present with probability , independently of all other potential edges. Likewise, let be the random -uniform hypergraph on , where edges are chosen uniformly at random among all sets of potential edges. The model can be gainfully viewed as a snapshot of the random hypergraph process where is obtained from by inserting an extra edge chosen uniformly at random among all remaining potential edges. For , these models are the typical random graph models, , and
As customary, we say that for a given ( resp.) some graph property holds with high probability, denoted w.h.p., if the probability that ( resp.) has property tends to 1 as . Further, is the sharp threshold for if for each (fixed), w.h.p. does not have and w.h.p does have . For the random hypergraph process, the stopping time of , denoted is the first moment that the process has this property ; we denote the hypergraph process stopped at this time by .
In one of the first papers on random graphs, Erdős and Rényi  showed that is a sharp threshold for connectivity in . Later, Stepanov  established the sharp threshold of connectivity for among other results. More recently, Bollobás and Thomason  proved the stronger result for the random graph process that w.h.p. the moment the graph process loses its last isolated vertex is also the moment that the process becomes connected; in other words, w.h.p. . We prove the analogous result for the the random -uniform hypergraph process; a consequence of this result is that is a sharp threshold of connectivity for .
There are various measures for the strength of connectedness of a connected graph, but here we will focus on -(vertex-)connectivity. For , a hypergraph with more than vertices is -connected if whenever vertices are deleted, along with their incident edges, the remaining hypergraph is connected. Note that the definition of 1-connectedness coincides with connectedness. Necessarily, for a hypergraph to be -connected, each vertex must have degree at least , because if a vertex has degree less than , then we can delete a neighbor from each incident edge to isolate . However, as commonly seen in these types of results, the main barrier to -connectivity in these random graph models arises from such vertices that can be separated from the rest of the graph by the deletion of their neighbors (see for instance Erdős-Rényi , Ivchenko , Bollobás ,). Here, we extend this idea to random -uniform hypergraphs; in particular, we find that if and , , then w.h.p. is not -connected and w.h.p. is -connected; also we find an analogous threshold value for .
A stronger result concerns the random graph process where edges are added one after another. Let and ; note that . In , Bollobás and Thomason showed that for (the graph case) and any , . We extend this result for -uniform random hypergraphs albeit for fixed and .
W.h.p, at the moment the -uniform hypergraph process loses its last vertex with degree less than , this process becomes -connected. Formally, for and (both fixed), .
To prove this result, we begin by determining the likely range of , and further that just prior to this window, at some edges, w.h.p. there are not many vertices of degree less than . Then, we prove that w.h.p. is almost -connected in the sense that whenever vertices are deleted, there is a massive component using almost all leftover vertices. Third, we show that w.h.p. to isolate a vertex of you would have to delete at least of its neighbors (this is trivially true for graphs, but not so for ). In particular, we show that w.h.p. edges incident to degree vertices have trivial intersection (just the vertex itself). Finally, we show that the probability that , but these three previous likely events also hold tends to zero, which completes the proof of the theorem. The following corollary is nearly immediate in light of the theorem.
(i) Let , where . W.h.p. is -connected, but not -connected. Further, the probability that is -connected tends to .
(ii) Let , where . W.h.p. is -connected, but not -connected. Further, the probability that is -connected tends to .
For the remainder of this paper, let and be fixed numbers.
2 Likely range of
Let , but , and . Then w.h.p.,
(i) the minimum degree of is and the number of vertices with degree is in the interval
(ii) there are no vertices of degree in .
Consequently, w.h.p. .
We prove that the number of vertices with degree , denoted by , is in the interval (2.1) by Chebyshev’s Inequality. Note that a given vertex can be in possible edges, so
Here and elsewhere in this paper, we use the identity , where and later, we use the inequality . Now
This latter fraction can be sharply approximated.
For the second factorial moment, we have that
We break this latter probability over , the number of edges that include both vertices and . In particular, vertex 1 is in edges that do not contain vertex and vice versa; further there are edges that include neither vertex 1 or 2. Since there are potential hyperedges containing both vertices and potential hyperedges containing one vertex but not the other, we have that
Just as in (2), we can estimate this latter fraction
Using these asymptotics, one can show that the ’th term in (2.3) is on the order of . In particular, the sum of the terms over , is . Therefore
By Chebyshev’s Inequality, is concentrated around its mean and in particular, w.h.p. is in the interval (2.1). To finish the proof of part (i), it remains to show that w.h.p. there are no vertices of degree less than , which can be done by a first moment argument using similar techniques to the asymptotics of . Similarly, for part (ii), one can easily show that the expected number of vertices of degree in tends to zero as well. ∎
3 is almost -connected
Now we will establish that w.h.p. is almost -connected in the sense that if vertices are deleted, then there remains a massive component containing almost all left-over vertices. To this end, we prove an analogous statement for the random Bernoulli hypergraph and use a standard conversion lemma to obtain the desired result for . In this next lemma, we pick a specific version of , one where .
Let and . With high probability,
(i) has the property “whichever vertices are deleted, there remains a giant component which includes all but up to leftover vertices.”
(ii) has the property “whichever vertices are deleted, there remains a giant component which includes all but up to leftover vertices.”
(i) Given a set of vertices, , let be the event that if the vertices are deleted from along with their incident edges, then there remains no components of size at least . In particular, we wish to show that w.h.p. is not in for any . Using the union bound over all element sets of as well as symmetry, we find that
where . Note that the remaining hypergraph left after deleting from is distributed as (this is the primary reason that we consider the Bernoulli hypergraph rather than ). Therefore is precisely the probability that does not have a component of size at least .
To bound , we note that any hypergraph on vertices without a component of size at least has a set of vertices such that there are no edges between and where To see this fact, consider a hypergraph on vertices without such a large component and let be the vertex sets of the components of in increasing order by their cardinalities. Then there is some minimal so that
Further, is not empty and since , we have that and
Clearly, there are no edges including a vertex of and .
and by symmetry over all such vertex sets ,
Further, this latter probability is symmetric about (i.e. the probabilities corresponding to and are equal). Hence
The number of potential edges that contain at least one vertex from and at least one vertex from is . Hence
where and are the sums over and respectively. We begin with analyzing since these bounds will be cruder and simpler.
Now let’s take on these binomial coefficient terms. Trivially and so
Further, the function is decreasing for , so we have that
Now our bound in (3.2) becomes
In the previous expression, the leading order terms in the first and third terms will cancel and the middle term is absorbed in the error. Namely, we have that
Now let’s take on the sum . We begin with taking the leading order terms in the exponent.
Uniformly over , we have that
Consequently, the exponent in (3.4) is
whence there is some (fixed) such that for all ,
However, for , we have that
This sum is dominated by the first term () because ratios of consecutive terms uniformly tend to zero. Consequently, we have that
which most definitely is and so by the bound (3.1), part (i) of the lemma is proved.
Part (ii) is established by using a standard conversion technique between and . For any hypergraph property , we have that
where is the number of edges of . Therefore, for any (possible) ,
For and , one can show that
Hence in our case,
In the proof of part (i), we found that this latter probability tends to zero superpolynomially fast. ∎
4 Quasi-disjoint Edges
For the random graph process (), it was found that the main barrier to -connectivity is the presence of vertices of degree less than , which could be isolated with the deletion of their neighbors (see Erdős-Rényi , Ivchenko , Bollobás ,). We will find a similar situation for the random hypergraph process.
However, we run into an additional issue here for hypergraphs. Even if the degree of a vertex is , we could isolate with the deletion of less than vertices. For instance, if all of ’s edges also include vertex , then the deletion of just from the hypergraph (along with its incident edges) will isolate from the rest of the hypergraph. Our ultimate goal in this section is to show that w.h.p. each vertex of has at least edges whose pairwise intersections are precisely ; in this case, for any vertex, you would need to delete at least of its neighbors to isolate it. To this end, we first prove that w.h.p. has this property for vertices with degree at least and as nearly as could be expected for vertices with degree .
A set of edges incident to vertex is quasi-disjoint if all pairwise intersections of these edges are formally, if , then
Let , where but . W.h.p., is such that
(i) the incident edges of a degree vertex form a quasi-disjoint set,
(ii) vertices with degree at least have a quasi-disjoint set of incident edges with size at least .
Note that both parts of this lemma are trivially true for and part (i) is also trivially true for . Let be the number of vertices whose maximum quasi-disjoint set has size and whose degree is . To prove this lemma, it suffices to show that w.h.p. for and , we have that , which is shown by a first moment argument. Now
where is the probability that a generic vertex has a maximum quasi-disjoint set of size and whose degree is . To bound this probability, note that has a set of quasi-disjoint edges and each of the remaining edges must have at least one vertex from the neighbors from the quasi-disjoint edges; further the remaining edges do not include . Hence
We gave sharp asymptotics for the last fraction in (2). Here and throughout the rest of the paper, we will use for when the formula for becomes too bulky. Therefore
for (independent of and ). Thus
which completes the proof of the lemma. ∎
W.h.p. each vertex of has a quasi-disjoint set of incident edges with size at least .
This lemma is trivially true for . Suppose that . Let be the event that has a vertex that does not have a quasi-disjoint set of edges with size at least ; we wish to show that . Let be as defined in Lemma 2.1. We have proved that w.h.p. and that does not have vertices of degree less than . Further, w.h.p. the number of degree vertices in is less than (Lemma 2.1). In addition, w.h.p. has the two properties of the previous lemma (Lemma 4.1). Let be the intersection of these four likely events. To prove the lemma, it suffices to show that .
Let be the vertex set of vertices of degree in . Note that
On the event that and occur and , necessarily some edge is added in the hypergraph process at some step such that includes both a vertex