probability that a random graph is connected

For a degree sequence $(d_1,\ldots, d_n)$ with $\min d_i \geq k$ pick a graph with that degree sequence uniformly at random. A random r-regular graph is almost surely r-connected; thus, maximally connected; A random r-regular graph has diameter at most d, where d has an upper bound of Θ(log r−1 (nlogn)) [1] The connectivity of a graph is an important measure of its resilience as a network. <> endobj Figure 4.5: Probability of connectivity with n=10 and a range of p. The vertical line shows the predicted critical value.¶, Figure 4.6: Probability of connectivity for several values of n and a range of p.¶. [1] B. Bollobás & W. Fernandez de la Vega, The diameter of random regular graphs, Combinatorica 2, 125–134 (1982). x��UMo1��W�hW��ߜPU��"�Cդ �$��U!�6Y=��7of��\+p\�gk��4�{vˌ׼��u��xW��0��swU��-��;aZ��MST)��O��f�s˜2!�8�k�Ś��l�Ϊ|�d��ð1��}�q;�S�!R�9�s9ec4Vlv0�,��)��r��Ъ��w�j��g��ސ�At��U��nXv9�1��x��"�$E�o)��{��0� D[{W��l NY��J�F. , n. This result extends Ross's recent theorems on connectivity of random graphs. The simulation runs in steps, and at each iteration active vertices propagate their newly produced/received data to their neighbors. {QE�&Q8^�h�r.� V�@ In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected. Usage data cannot currently be displayed. In other words, if Gis a random graph with In the G(n, M) model, a graph is chosen uniformly at random from the collection of all graphs which have n nodes and M edges. These experimental results are consistent with the analytic results Erdős and Rényi presented in their papers. and – The construction of the G(N, p) model is closer to the way real systems develop. np.mean is a NumPy function that computes the mean of this list, treating True as 1 and False as 0. Published online by Cambridge University Press: URL: /core/journals/journal-of-applied-probability. The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. The probability that the hyperbolic random graph is connected Michel Bode Nikolaos Fountoulakis Tobias Muller y March 6, 2014 Abstract This work is a study of a family of random geometric graphs on the hyperbolic plane. In this post I analyze a network topology based on unweighted random regular graphs, and evaluate its robustness for data replication amid partial network disruption. We can get a clearer view of the transition by estimating the probability of connectivity for a range of values of $p$: The NumPy function logspace returns an array of $11$ values from $10^{−2.5}$ to $10^0 = 1$, equally spaced on a logarithmic scale. The first one is 2-regular (two edges per vertex) and the following two are 3-regular (three edges per vertex). It’s worth noting that the inferred diameter standard deviation started reasonably small (less than 0.5) and diminished to insignificant values as the graph size and degree increased. Nonetheless, as p nears 1.0, we are certain to end up with at least one disconnected vertex, meaning that we won’t be able to assess a valid number of simulation iterations for the replication to reach the entire graph. Graph theory is extensively studied, experimented on and applied to communications networks . Depending on a communication network’s requirements it may benefit from adopting one or another network topology: Point to point, Ring, Star, Tree, Mesh, and so forth. Figure 4.5: Probability of connectivity with n=10 and a range of p. The vertical line shows the predicted critical value. The result, tf, is a list of boolean values: True for each graph that’s connected and False for each one that’s not. There is a collection of conjectured by Jeff Kahn and me trying to suggest a very general connection of this type. These conjectures are presented in the pape The algorithm used in this code sample for calculating the edge connectivity between two vertices is straightforward, but a bit more extensive. This data will be updated every 24 hours. It’s important to highlight that as the probability of cutting off graph vertices increases, the number of simulation iterations required for reaching the totality of graph vertices becomes more volatile since, the way my code was devised, a new random regular graph is sampled and the current disruption level is randomly applied at each retrial. Probability that a random graph is fully connected. A widely known algorithm for generating random regular graphs is the pairing model, which was first given by Bollobas2. %�� This simple model has proven networks properties and is a good baseline to compare real-world graph properties with. The result is the fraction of random graphs that are connected. Figure 4.6 shows similar results for larger values of n. As n increases, the critical value gets smaller and the transition gets more abrupt. /Filter /FlateDecode %�쏢 x��XK��6��W�(#�D*��Y�!��q��Φ��)r(��79 6hr8~��|,[&T��_,n�g��|�/>\�%��u#��B�v��ǽB��V�Փ�uѲNY%�;�۷]y�2; |>��댓./n�g �-�d�(gWY6�-{Ô*�m�-g�śꧺ��6uÙ4��O��Z�jK$K��pQ7�)��-\C�,6��o��3��Wv�G^�3�!W`*Mτ��ucl[�P7EUYs[]��p=Y�i� U�w�]�g�qe�?��S��V��#��ފ��B֎�?7�[4�j`��x/�G l�Nw��z�Dn��s�k��WQXǲ�"�� 0��r�h�u�D�qƃqo�sH0�3�(��C~�mAdh� pI��Mﷃ�)��(�t�81� �c��#�S��U9�]r=��Y��Nca��r)�4v��$K�(kЍS��X8��E~�s$_�?��%�N�с�2\ڑ�"?��Fu+C�;��H��Ķ��,\Q��W��n�阖�/�bR�ڔf\ A regular graph is a graph where each vertex has the same number of neighbors; i.e. endobj Here’s the method that I implemented for running it for any given graph size and degree: The results for running it with parameters n = 1000 and r = [4, 8, 12] are given in the chart that follows: We can verify that the larger the graph’s degree, the less significant the effects of disruption levels are, which makes sense since intuitively there are much more path options available for the information to replicate. Verify if the resulting graph is simple, i.e., make sure that none of the vertices have loops (self-connections) or multiple edges (more than one connection to the same vertex). The simulation runs until 100% of vertices successfully receive data originated from a source vertex chosen arbitrarily, or until the number of iterations becomes greater than the number of vertices, indicating that the level of disruption has effectively separated the graph into two isolated subgraphs. /Length 2479 Full text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views. For the lowest graph degree tested with [n=1000, r=4], however, the effects of disruption were quite noticeable, spiking to a total of 42 iterations for 25% disruption and not being able to reach the entirety of vertices for disruption levels above that (that’s why it’s not even plotted in the chart). /Filter /FlateDecode ��D�w� U¤:0*q��m�!�;��Tx�6�|S�U�R&|��w��;�� Wk��Z��7��i�i,�}��x�:�hN��ݏ��R�,IIi��: �Qh)Y�� ~��pz�8�ے �8�"��t�k��Y *��_�H/q�Z��֟�OĨ��̬ޮR6��J�a�\��N1�*��l��*Ua��r�Ԇ��J� /u]�m$�؎>.q[��2f�x�B�5ʇ4��T��M߼M� ��1Hf I��H�vs�ڼ��>}��$�B��ÇL�Rv�~�~6�� e��3�n��X�O�$b�/�cٵ��J��?�H�4,ۍ{z|c��O��1��t�G��b� �s�u�i�M$G�0B��G ��Kj�p�o`��yfFy[2��f͹��I+�?&�U�`�͓c��{�7��@�3s�=Ɏ�l0�c�zI!u��z_�"��e�/��䦧KN�^n�%�T;~��#�y"*|��疳��]~*�;�(�Nz=�}n+L��u�Z[�q %ʟW tT_��X OX�g�6/pXk��Tj�6��,@t��|� n��n�� ;^`Pc��S�n��T�^��ς�H;s-I~pq��˱��8�"�DH��B��ۮjN9�D�� q��[�LZ鵯�M}��nb� S��*w��ly ��ks�q ;6�. Mathematicians have been studying random graphs for a long time, starting with the work by Paul ErD os and Alfr ed R enyi. Even though 0-regular (disconnected), 1-regular (two vertices connected by single edge) and 2-regular (circular) graphs take only one form each, r-regular graphs of the third degree and upwards take multiple distinct forms by combining their vertices in a variety of different ways. We can estimate it by generating a large number of random graphs and counting how many are connected. According to Erdős and Rényi, $p* = lnn / n = 0.23$. ��D�w� U¤:0*q��m�!�;��Tx�6�|S�U�R&|��w��;�� Wk��Z��7��i�i,�}��x�:�hN��ݏ��R�,IIi��: �Qh)Y�� ~��pz�8�ے �8�"��t�k��Y *��_�H/q�Z��֟�OĨ��̬ޮR6��J�a�\��N1�*��l��*Ua��r�Ԇ��J� /u]�m$�؎>.q[��2f�x�B�5ʇ4��T��M߼M� ��1Hf I��H�vs�ڼ��>}��$�B��ÇL�Rv�~�~6�� e��3�n��X�O�$b�/�cٵ��J��?�H�4,ۍ{z|c��O��1��t�G��b� �s�u�i�M$G�0B��G ��Kj�p�o`��yfFy[2��f͹��I+�?&�U�`�͓c��{�7��@�3s�=Ɏ�l0�c�zI!u��z_�"��e�/��䦧KN�^n�%�T;~��#�y"*|��疳��]~*�;�(�Nz=�}n+L��u�Z[�q %ʟW tT_��X OX�g�6/pXk��Tj�6��,@t��|� n��n�� ;^`Pc��S�n��T�^��ς�H;s-I~pq��˱��8�"�DH��B��ۮjN9�D�� q��[�LZ鵯�M}��nb� S��*w��ly ��ks�q ;6�. It’s a simple algorithm to implement and works fast enough for small degrees, but it becomes slow when the degree is large, because with high probability we will get a multigraph with loops and multiple edges, so we have to abandon this multigraph and start the algorithm again3. 5 0 obj %PDF-1.5 Email your librarian or administrator to recommend adding this journal to your organisation's collection. 3 0 obj << We chose 0.23 because it is close to the critical value where the probability of connectivity goes from near 0 to near 1. Also relevant for communication networks is the graph diameter, which is the greatest distance between any pair of vertices, and hence is qualitatively related to the complexity of messaging routing within the network. [2] B. Bollobás, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, Preprint Series, Matematisk Institut, Aarhus Universitet (1979). %PDF-1.3 For each value of p in the array, we compute the probability that a graph with parameter p is connected and store the results in ys. The image below shows a few examples: These sample graphs are regular since we can confirm that every vertex has exactly the same number of edges. The Erdös-Rényi Random Graph Model is the simplest model of graphs. In the simulation ran with parameters [n=1000, r=12] it was only required two additional iterations (6 in total) at 50% disruption for the source data to reach all graph vertices when compared with the base case in which all vertices were active. Supporting source code for this article can be found in this GitHub repository. (�kH-C� �UU.6˪��f(�L��8a��˦Hendstream Otherwise finish. xڥYK��6��HEt�H 6v6��L؎q�7|ρ��*v��MQ�ӻ��RP��=�� !��T�/� Collapse the points, so that each bucket (and thus the points it contains) maps onto a single vertex of the original graph. 3 0 obj << �-;�,�c?0ɤt�-L�{�˲�׽d��D��o��t��@��-��ES��Ee��B��I��h}ϒ��"�3�w��̥ЎJq8)�C�.��i��һ��U�Y��l��S�?/�R :�? So after implementing this method I ran it against random regular graphs of varying sizes and degrees. %PDF-1.5 every vertex has the same degree or valency. If a valid path is found increment the counter, remove all path edges from the graph and go back to step. For example, in the G(3, 2) model, each of the three possible graphs on three vertices and two edges are included with probability 1/3. Probability that a random graph is fully connected. We chose 0.23 because it is close to the critical value where the probability of connectivity goes from near 0 to near 1.

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