On realizations of a joint degree matrix

Czabarka, Éva; Dutle, Aaron; Erdős, Péter L.; Miklós, István

doi:10.1016/j.dam.2014.10.012

Cited by 28 publications

(68 citation statements)

References 9 publications

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“…• With probability 1/2, do nothing. This chain is irreducible, aperiodic and symmetric [1,11]. Like the switch chain defined above, P(G, G ′ ) −1 ≤ n 4 for all adjacent G, G ′ ∈ G ′ (c, d), and also the maximum in-and out-degrees of the state space graph are less than n 4 .…”

Section: Joint Degree Matrix Modelmentioning

confidence: 95%

“…, 14}. A traversal of P is given by the sequence (0, 6), (1, 7), (0, 8), (1,9), (2, 10), (3,11), (4,12), (5,13), (6,14).…”

Section: D11 Canonical Pathsmentioning

confidence: 99%

“…Now, let G ∈ G ′ (c, d) for some tuple (c ′ , d ′ ) as in the definition of G ′ (c, d) at the start of Section D. We show that with at most four hinge flip operations, we can transform G into a perturbed auxiliary state G * ∈ G ′ (c, d) for which its tuple (c * , d * ) is edge-balanced. 11 . Assume without loss of generality that we are in the first case.…”

Section: E Strongly Stable Families For the Pam Modelmentioning

confidence: 99%

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Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices

Amanatidis¹,

Kleer²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The switch Markov chain has been extensively studied as the most natural Markov Chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We use comparison arguments with other-less natural but simpler to analyze-Markov chains, to show that the switch chain mixes rapidly in two different settings. We first study the classic problem of uniformly sampling simple undirected, as well as bipartite, graphs with a given degree sequence. We apply an embedding argument, involving a Markov chain defined by Jerrum and Sinclair (TCS, 1990) for sampling graphs that almost have a given degree sequence, to show rapid mixing for degree sequences satisfying strong stability, a notion closely related to P-stability. This results in a much shorter proof that unifies the currently known rapid mixing results of the switch chain and extends them up to sharp characterizations of P-stability. In particular, our work resolves an open problem posed by Greenhill (SODA, 2015).Secondly, in order to illustrate the power of our approach, we study the problem of uniformly sampling graphs for which-in addition to the degree sequence-a joint degree distribution is given. Although the problem was formalized over a decade ago, and despite its practical significance in generating synthetic network topologies, small progress has been made on the random sampling of such graphs. The case of a single degree class reduces to sampling of regular graphs, but beyond this almost nothing is known. We fully resolve the case of two degree classes, by showing that the switch Markov chain is always rapidly mixing. Again, we first analyze an auxiliary chain for strongly stable instances on an augmented state space and then use an embedding argument.

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Section: Joint Degree Matrix Modelmentioning

confidence: 95%

“…, 14}. A traversal of P is given by the sequence (0, 6), (1, 7), (0, 8), (1,9), (2, 10), (3,11), (4,12), (5,13), (6,14).…”

Section: D11 Canonical Pathsmentioning

confidence: 99%

Section: E Strongly Stable Families For the Pam Modelmentioning

confidence: 99%

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Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices

Amanatidis¹,

Kleer²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The problem of graphicality for JDMs asks whether a specified symmetric matrix can be the JDM of a simple graph. Our starting point is an Erdős-Gallai-like theorem that gives the requirements for a JDM to be graphical [48][49][50].…”

Section: Mathematical Foundations 21 Graphicality Of Jdmsmentioning

confidence: 99%

Exact sampling of graphs with prescribed degree correlations

et al. 2015

Self Cite

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Many real-world networks exhibit correlations between the node degrees. For instance, in social networks nodes tend to connect to nodes of similar degree and conversely, in biological and technological networks, high-degree nodes tend to be linked with low-degree nodes. Degree correlations also affect the dynamics of processes supported by a network structure, such as the spread of opinions or epidemics. The proper modelling of these systems, i.e., without uncontrolled biases, requires the sampling of networks with a specified set of constraints. We present a solution to the sampling problem when the constraints imposed are the degree correlations. In particular, we develop an exact method to construct and sample graphs with a specified joint-degree matrix, which is a matrix providing the number of edges between all the sets of nodes of a given degree, for all degrees, thus completely specifying all pairwise degree correlations, and additionally, the degree sequence itself. Our algorithm always produces independent samples without backtracking. The complexity of the graph construction algorithm is NMwhere N is the number of nodes and M is the number of edges.where d i − denotes the in-degree, and d i + the out-degree, of node i. The situation of most practical interest is when we demand the graph with a given degree sequence to be a simple graph, which has the additional constraints that there can be at most one link (in each direction, if directed) between any two nodes, and that no link starts and ends on the same node (no self-loops). However, not all positive integer sequences can serve as the sequence of the degrees of some simple graph. If such a graph does exist, then the sequence is said to be graphical. Any simple graph (just 'graph' from here on) with the prescribed node degrees is said to realize the degree sequence, and it is called a graphical realization of the sequence. The two main results used to test the graphicality of an undirected degree sequence are the Erdős-Gallai theorem [5] and the Havel-Hakimi theorem [6,7]. For directed networks, instead, the main theorem characterizing the graphicality of a BDS is due to Fulkerson [8]. More recently, exploiting a formulation based on recurrence relations, new methods were introduced to implement these tests with a worst case computational complexity that is only linear in the number of nodes [9][10][11]. The advantage of these methods over others with similar complexity [12] is that they also allow a straightforward algorithmic implementation.While the above results provide complete and practical answers to the question of the graphicality of sequences of integers, they do not suffice to solve the problem of constructing graphs with prescribed degrees. One of the main issues with constructing graphs for the purpose of ensemble modeling is that, except for networks of just a few nodes, the number of graphs realizing a degree sequence, or other possible constraints, is generally so large that their complete enumeration is impractical. Therefore, one has...

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“…This information is quantified in the symmetric joint-degree matrix (JDM) whose ( i, j ) element is the number of edges between nodes of degree i and nodes of degree j [31]. The necessary and sufficient condition for a simple network to exist for a given JDM is given by the Erdős–Gallai type theorem [1, 8, 31]:…”

Section: Introductionmentioning

confidence: 99%

Generating bipartite networks with a prescribed joint degree distribution

Boroojeni

Dewar

et al. 2017

Journal of Complex Networks

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We describe a class of new algorithms to construct bipartite networks that preserves a prescribed degree and joint-degree (degree-degree) distribution of the nodes. Bipartite networks are graphs that can represent real-world interactions between two disjoint sets, such as actor-movie networks, author-article networks, co-occurrence networks, and heterosexual partnership networks. Often there is a strong correlation between the degree of a node and the degrees of the neighbors of that node that must be preserved when generating a network that reflects the structure of the underling system. Our bipartite 2K (B2K) algorithms generate an ensemble of networks that preserve prescribed degree sequences for the two disjoint set of nodes in the bipartite network, and the joint-degree distribution that is the distribution of the degrees of all neighbors of nodes with the same degree. We illustrate the effectiveness of the algorithms on a romance network using the NetworkX software environment to compare other properties of a target network that are not directly enforced by the B2K algorithms. We observe that when average degree of nodes is low, as is the case for romance and heterosexual partnership networks, then the B2K networks tend to preserve additional properties, such as the cluster coefficients, than algorithms that do not preserve the joint-degree distribution of the original network.

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On realizations of a joint degree matrix

Cited by 28 publications

References 9 publications

Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices

Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices

Exact sampling of graphs with prescribed degree correlations

Generating bipartite networks with a prescribed joint degree distribution

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